A354685 - OEIS

A354685

a(n) = n! * Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * H(k), where H(k) is the k-th harmonic number.

0, 1, 5, 50, 854, 22354, 833244, 41974176, 2748169584, 226916044848, 23069499189120, 2831994888419520, 413051278946186880, 70608112721914654080, 13982696139441640584960, 3175762393024883382067200, 820007850688478572529203200, 238863690100874514528150681600

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Vaclav Kotesovec, Graph - the asymptotic ratio

FORMULA

Sum_{n>=0} a(n) * x^n / n!^2 = Sum_{n>=1} H(n) * (-log(1-x))^n / n!.

a(n) ~ n!^2 * (log(log(n)) + gamma + 1/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 03 2022

MATHEMATICA

Table[n! Sum[(-1)^(n - k) StirlingS1[n, k] HarmonicNumber[k], {k, 1, n}], {n, 0, 17}]

nmax = 17; CoefficientList[Series[Sum[HarmonicNumber[k] (-Log[1 - x])^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2